Optimal. Leaf size=78 \[ \frac{b^2 (3 a+2 b) \tanh (c+d x)}{d}+\frac{(a+b)^3 \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac{1}{2} x (a-5 b) (a+b)^2+\frac{b^3 \tanh ^3(c+d x)}{3 d} \]
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Rubi [A] time = 0.089681, antiderivative size = 78, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.174, Rules used = {3675, 390, 385, 206} \[ \frac{b^2 (3 a+2 b) \tanh (c+d x)}{d}+\frac{(a+b)^3 \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac{1}{2} x (a-5 b) (a+b)^2+\frac{b^3 \tanh ^3(c+d x)}{3 d} \]
Antiderivative was successfully verified.
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Rule 3675
Rule 390
Rule 385
Rule 206
Rubi steps
\begin{align*} \int \cosh ^2(c+d x) \left (a+b \tanh ^2(c+d x)\right )^3 \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\left (a+b x^2\right )^3}{\left (1-x^2\right )^2} \, dx,x,\tanh (c+d x)\right )}{d}\\ &=\frac{\operatorname{Subst}\left (\int \left (b^2 (3 a+2 b)+b^3 x^2+\frac{(a-2 b) (a+b)^2+3 b (a+b)^2 x^2}{\left (1-x^2\right )^2}\right ) \, dx,x,\tanh (c+d x)\right )}{d}\\ &=\frac{b^2 (3 a+2 b) \tanh (c+d x)}{d}+\frac{b^3 \tanh ^3(c+d x)}{3 d}+\frac{\operatorname{Subst}\left (\int \frac{(a-2 b) (a+b)^2+3 b (a+b)^2 x^2}{\left (1-x^2\right )^2} \, dx,x,\tanh (c+d x)\right )}{d}\\ &=\frac{(a+b)^3 \cosh (c+d x) \sinh (c+d x)}{2 d}+\frac{b^2 (3 a+2 b) \tanh (c+d x)}{d}+\frac{b^3 \tanh ^3(c+d x)}{3 d}+\frac{\left ((a-5 b) (a+b)^2\right ) \operatorname{Subst}\left (\int \frac{1}{1-x^2} \, dx,x,\tanh (c+d x)\right )}{2 d}\\ &=\frac{1}{2} (a-5 b) (a+b)^2 x+\frac{(a+b)^3 \cosh (c+d x) \sinh (c+d x)}{2 d}+\frac{b^2 (3 a+2 b) \tanh (c+d x)}{d}+\frac{b^3 \tanh ^3(c+d x)}{3 d}\\ \end{align*}
Mathematica [A] time = 0.850207, size = 69, normalized size = 0.88 \[ \frac{4 b^2 \tanh (c+d x) \left (9 a-b \text{sech}^2(c+d x)+7 b\right )+6 (a-5 b) (a+b)^2 (c+d x)+3 (a+b)^3 \sinh (2 (c+d x))}{12 d} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.047, size = 148, normalized size = 1.9 \begin{align*}{\frac{1}{d} \left ({a}^{3} \left ({\frac{\cosh \left ( dx+c \right ) \sinh \left ( dx+c \right ) }{2}}+{\frac{dx}{2}}+{\frac{c}{2}} \right ) +3\,{a}^{2}b \left ( 1/2\,\cosh \left ( dx+c \right ) \sinh \left ( dx+c \right ) -1/2\,dx-c/2 \right ) +3\,a{b}^{2} \left ( 1/2\,{\frac{ \left ( \sinh \left ( dx+c \right ) \right ) ^{3}}{\cosh \left ( dx+c \right ) }}-3/2\,dx-3/2\,c+3/2\,\tanh \left ( dx+c \right ) \right ) +{b}^{3} \left ({\frac{ \left ( \sinh \left ( dx+c \right ) \right ) ^{5}}{2\, \left ( \cosh \left ( dx+c \right ) \right ) ^{3}}}-{\frac{5\,dx}{2}}-{\frac{5\,c}{2}}+{\frac{5\,\tanh \left ( dx+c \right ) }{2}}+{\frac{5\, \left ( \tanh \left ( dx+c \right ) \right ) ^{3}}{6}} \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.15486, size = 346, normalized size = 4.44 \begin{align*} \frac{1}{8} \, a^{3}{\left (4 \, x + \frac{e^{\left (2 \, d x + 2 \, c\right )}}{d} - \frac{e^{\left (-2 \, d x - 2 \, c\right )}}{d}\right )} - \frac{3}{8} \, a^{2} b{\left (4 \, x - \frac{e^{\left (2 \, d x + 2 \, c\right )}}{d} + \frac{e^{\left (-2 \, d x - 2 \, c\right )}}{d}\right )} - \frac{1}{24} \, b^{3}{\left (\frac{60 \,{\left (d x + c\right )}}{d} + \frac{3 \, e^{\left (-2 \, d x - 2 \, c\right )}}{d} - \frac{121 \, e^{\left (-2 \, d x - 2 \, c\right )} + 201 \, e^{\left (-4 \, d x - 4 \, c\right )} + 147 \, e^{\left (-6 \, d x - 6 \, c\right )} + 3}{d{\left (e^{\left (-2 \, d x - 2 \, c\right )} + 3 \, e^{\left (-4 \, d x - 4 \, c\right )} + 3 \, e^{\left (-6 \, d x - 6 \, c\right )} + e^{\left (-8 \, d x - 8 \, c\right )}\right )}}\right )} - \frac{3}{8} \, a b^{2}{\left (\frac{12 \,{\left (d x + c\right )}}{d} + \frac{e^{\left (-2 \, d x - 2 \, c\right )}}{d} - \frac{17 \, e^{\left (-2 \, d x - 2 \, c\right )} + 1}{d{\left (e^{\left (-2 \, d x - 2 \, c\right )} + e^{\left (-4 \, d x - 4 \, c\right )}\right )}}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.0419, size = 903, normalized size = 11.58 \begin{align*} \frac{3 \,{\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )} \sinh \left (d x + c\right )^{5} - 4 \,{\left (18 \, a b^{2} + 14 \, b^{3} - 3 \,{\left (a^{3} - 3 \, a^{2} b - 9 \, a b^{2} - 5 \, b^{3}\right )} d x\right )} \cosh \left (d x + c\right )^{3} - 12 \,{\left (18 \, a b^{2} + 14 \, b^{3} - 3 \,{\left (a^{3} - 3 \, a^{2} b - 9 \, a b^{2} - 5 \, b^{3}\right )} d x\right )} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{2} +{\left (9 \, a^{3} + 27 \, a^{2} b + 99 \, a b^{2} + 65 \, b^{3} + 30 \,{\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )} \cosh \left (d x + c\right )^{2}\right )} \sinh \left (d x + c\right )^{3} - 12 \,{\left (18 \, a b^{2} + 14 \, b^{3} - 3 \,{\left (a^{3} - 3 \, a^{2} b - 9 \, a b^{2} - 5 \, b^{3}\right )} d x\right )} \cosh \left (d x + c\right ) + 3 \,{\left (5 \,{\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )} \cosh \left (d x + c\right )^{4} + 2 \, a^{3} + 6 \, a^{2} b + 30 \, a b^{2} + 10 \, b^{3} +{\left (9 \, a^{3} + 27 \, a^{2} b + 99 \, a b^{2} + 65 \, b^{3}\right )} \cosh \left (d x + c\right )^{2}\right )} \sinh \left (d x + c\right )}{24 \,{\left (d \cosh \left (d x + c\right )^{3} + 3 \, d \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{2} + 3 \, d \cosh \left (d x + c\right )\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 2.06105, size = 360, normalized size = 4.62 \begin{align*} \frac{12 \,{\left (a^{3} - 3 \, a^{2} b - 9 \, a b^{2} - 5 \, b^{3}\right )} d x - 3 \,{\left (2 \, a^{3} e^{\left (2 \, d x + 2 \, c\right )} - 6 \, a^{2} b e^{\left (2 \, d x + 2 \, c\right )} - 18 \, a b^{2} e^{\left (2 \, d x + 2 \, c\right )} - 10 \, b^{3} e^{\left (2 \, d x + 2 \, c\right )} + a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )} e^{\left (-2 \, d x - 2 \, c\right )} + 3 \,{\left (a^{3} e^{\left (2 \, d x + 12 \, c\right )} + 3 \, a^{2} b e^{\left (2 \, d x + 12 \, c\right )} + 3 \, a b^{2} e^{\left (2 \, d x + 12 \, c\right )} + b^{3} e^{\left (2 \, d x + 12 \, c\right )}\right )} e^{\left (-10 \, c\right )} - \frac{16 \,{\left (9 \, a b^{2} e^{\left (4 \, d x + 4 \, c\right )} + 9 \, b^{3} e^{\left (4 \, d x + 4 \, c\right )} + 18 \, a b^{2} e^{\left (2 \, d x + 2 \, c\right )} + 12 \, b^{3} e^{\left (2 \, d x + 2 \, c\right )} + 9 \, a b^{2} + 7 \, b^{3}\right )}}{{\left (e^{\left (2 \, d x + 2 \, c\right )} + 1\right )}^{3}}}{24 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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